Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 33 / 6

Introduce a latent indicator $Z_i=1$ for a structural zero. In this parametrization $P(Z_i=1)={\pi }_i$, and conditional on $Z_i=0$ the response has a Poisson distribution with mean ${\mu }_i$. The zero-inflated Poisson regression usesIts observed probability mass function is ${\pi }_i+(1-{\pi }_i)e^{-{\mu }_i}$ at zero and $(1-{\pi }_i)e^{-{\mu }_i}{\mu }_i^{y_i}/y_i!$ at a positive count. The zero-mixing probability here is explicitly ${\pi }_i$; it is distinct from the weight of the count component.

For the expectation-maximization algorithm, start with positive ${\mu }_i$ and $0<{\pi }_i<1$. At iteration $k$, the E-step computesThis distinguishes structural zeros from Poisson-generated zeros. Up to terms independent of the new parameters, the expected complete-data log-likelihood isIt separates into a fractional-response logistic fit and a weighted Poisson fit. Because treatment is binary, each component is saturated over its two treatment groups, so the M-step has closed forms.

For $j=0,1$, let $I_j=\{i:x_i=j\}$, $n_j=|I_j|$, $S_j={\sum }_{i\in I_j}{\tau }_i$, and $C_j={\sum }_{i\in I_j}{y}_i$. The M-step score equations giveHere $\sum (1-{\tau }_i)y_i=C_j$, since every positive count has ${\tau }_i=0$. The explicit coefficient updates areIterate the E- and M-steps until the observed log-likelihood and parameter estimates stabilize. This is the EM algorithm for zero-inflated Poisson regression with explicit binary-group updates; merely naming two regression routines would not supply these expressions. Both treatment groups must be represented for both contrasts to be identifiable. Zero fitted group means or endpoint mixing probabilities are boundary solutions, interpreted through limits of the log or logit coefficients. As with other mixture models, multiple starts help distinguish competing stationary solutions; EM increases the likelihood but does not guarantee a global maximum from an arbitrary start.

There are two different effect scales in this zero-inflated Poisson regression. The count component describes the mean $\mu$ conditional on belonging to the susceptible component, including its possible zero outcomes. Its logarithmic link function coefficients exponentiate to ratios of those conditional means. The zero component describes the structural-zero probability $\pi$; its logit coefficients exponentiate to odds ratios, not probability ratios.

Holding the other covariates fixed, Centre B versus Centre A changes the susceptible count mean by the factoran increase of about $54.4\%$. It simultaneously multiplies the odds of being a structural zero byThus Centre B is associated both with higher count intensity among susceptible patients and with greater odds of belonging to the never-at-risk class.

Use no personality disorder as the reference. For women, borderline disorder multiplies the susceptible count mean by $e^{-0.224673}=0.7988$, and other disorder multiplies it by $e^{-1.564576}=0.2092$. Because the count model has sex-by-disorder interaction terms, the corresponding male comparisons must add the interactions before exponentiating:These compare men with the stated disorder to otherwise comparable men without a disorder; the female comparisons use female reference patients. The interaction multipliers $0.7865$ and $4.3157$ alone are ratios of these sex-specific disorder ratios, not the overall male disorder effects.

In the zero component there is no sex-by-disorder interaction. Borderline and other disorders multiply the structural-zero odds relative to no disorder by $0.4217$ and $0.5504$, respectively, holding centre fixed; these comparisons are the same for both sexes in the fitted zero model.

The marginal mean is $E(Y)=(1-\pi )\mu$. Therefore none of these count multipliers alone is the population-average change in expected episodes when the same covariate also changes $\pi$. For a count coefficient change $b$ and zero-logit change $g$ from baseline zero predictor $\eta$, the marginal mean ratio isFor centre use $(b,g)=(0.434635,0.6715)$; for disorder use its sex-specific count contrast and its zero contrast. This is the component and marginal effects in zero-inflated regression distinction needed to interpret these results carefully.

Let $Z=1$ mean membership in the structural-zero component. Under the fitted zero-inflated Poisson regression, $P(Y=0\mid Z=1)=1$, whereas $P(Y=0\mid Z=0)=e^{-\mu }$. By Bayes theorem, the posterior structural-zero probability isUsing the printed predictions for this patient givesThe fitted conditional probability is about $50.0\%$. It is larger than the prior fitted structural-zero probability $0.4448758$, because observing no episodes increases the probability of latent membership in that component. The interpretation “never at risk” is the model's structural class; an observed six-month zero alone does not identify the class with certainty.